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The Journal of Chemical Physics

Convenient and accurate discretized path integral methods for equilibrium quantum mechanical calculations

Kenneth S. Schweizer, Richard M. Stratt, David Chandler, Peter Guy Wolynes

Published Aug 1, 1981• 259 citations
Publisher

Abstract

In the path integral representation of quantum theory, a few body quantum problem becomes a classical many body problem. To exploit this isomorphism, it becomes necessary to develop methods by which degrees of freedom can be explicitly removed from consideration. The interactions among the remaining relevant variables are described by effective interactions. In this paper, we present a general methodology to carry out the reduction in numbers of degrees of freedom. Certain path integral algorithms are shown to correspond to reference systems for the full isomorphic classical many body problem. The correspondence allows one to determine systematic corrections to the algorithms by low order perturbation approximations familiar in the theory of simple classical fluids. We show how to use discretized path integrals to compute rigorous upper and lower bounds to the free energy for nontrivial quantum systems, and we discuss how to optimize the upper bounds with variational theories. Several illustrative examples are provided including treatments of a tunneling system that is strongly coupled to a nonadiabatic heat bath. The examples demonstrate that even for strongly perturbed systems in which the quantum energy scales are as large as 40kBT, the methods discussed herein provide a practical route for obtaining equilibrium properties with both high accuracy and analytical simplicity.

DOI
10.1063/1.442141

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In the path integral representation of quantum theory, a few body quantum problem becomes a classical many body problem. To exploit this isomorphism, it becomes necessary to develop methods by which degrees of freedom can be explicitly removed from consideration. The interactions among the remaining relevant variables are described by effective interactions. In this paper, we present a general methodology to carry out the reduction in numbers of degrees of freedom. Certain path integral algorithms are shown to correspond to reference systems for the full isomorphic classical many body problem. The correspondence allows one to determine systematic corrections to the algorithms by low order perturbation approximations familiar in the theory of simple classical fluids. We show how to use discretized path integrals to compute rigorous upper and lower bounds to the free energy for nontrivial quantum systems, and we discuss how to optimize the upper bounds with variational theories. Several illustrative examples are provided including treatments of a tunneling system that is strongly coupled to a nonadiabatic heat bath. The examples demonstrate that even for strongly perturbed systems in which the quantum energy scales are as large as 40kBT, the methods discussed herein provide a practical route for obtaining equilibrium properties with both high accuracy and analytical simplicity.
From a discretization of the path integral formulation of quantum mechanics, it is possible to relate equilibrium quantum many body theory to classical statistical mechanics. In this paper, we significantly extend and analyze this well known isomorphism in terms of the equilibrium theory of classical molecular fluids composed of flexible polyatomic species. We show how quantum influence functionals are isomorphic to classical cavity distribution functions. The former describe the influence of surrounding media on the dynamics of quantal degrees of freedom, and the latter describe environmental effects for classical models of flexible molecules and chemical equilibria. The connection allows the use of classical theories to perform nonperturbative calculations of influence functionals which treat the influence functionals and many body correlation functions in a self-consistent fashion. We illustrate the computational advantages of the method by studying its predictions for a hard sphere model of liquid helium above the l transition. The nature of quantum indistinguishability of identical particles (i.e., quantum exchange) is treated in our theory in terms of an exact isomorphism with chemical equilibria. This connection allows the treatment of exchange in condensed phases in terms of the classical law of mass action, and provides a computational advance over existing methods for interacting systems. By picturing exchange in terms of classical association equilibrium, we arrive at a view (hinted at long ago by Feynman and by Penrose and Onsager) in which the Bose condensation is related to an equilibrium polymeric sol–gel transition. Thus, below the l transition, the correlations in liquid helium are equivalent to those in a classical fluid containing a finite concentration of macroscopic polymers. We stress how the path integral aspect of the isomorphism leads to useful geometrical interpretations of quantum phenomena. For example, tunneling phenomena can be viewed in terms of solitonic (or instantonic) configurations or flexible chain molecules. With this picture, we show how the isomorphism can be employed to understand both adiabatic and nonadiabatic solvent effects on chemical bonding. For concreteness, we provide a detailed analysis of a particular model of the chemical bond for which a partial summation over intermediate quantum paths leads to an Ising model problem in the isomorphism. While applications like this are presented in the form of qualitative illustrations, a variety of methods can be employed to produce quantitative results. We sketch how these calculations can be performed for various problems, making connections with methods like the renormalization group (RG) technique. The classical isomorphism together with the modern theory of classical polyatomic systems provides a powerful framework for quantitative solutions of condensed matter quantum mechanical problems.
In the path integral representation of the thermal density matrix, the sum over quantum paths is equivalent to the configurational partition function sum for a system composed of long polyatomic molecules. We give an iterative (renormalization group) approach which successively reduces the number of degrees of freedom in this isomorphic classical many-body problem to a more tractable few-body problem. We give as examples a two state tunneling system coupled to an adiabatic and nonadiabatic bath and an impurity with spin coupled to conduction band electrons.
We present the path-integral approach to treat quantum phase transitions and critical phenomena for few-body quantum systems. Allowing the space and time variables to have discrete values, we turn the quantum problem into an effective classical lattice problem. Imposing the constraint that any change in space time must preserve the scaling invariance of Brownian paths, we show that the mapped classical lattice system has a known scaling behavior when the particle is free, which breaks down when the strength of the interaction potential reaches a certain value. In principle, any quantity with known scaling behavior may be used to determine the transition point. We illustrate the method by numerically evaluating the correlation length and the radial mean distance for a system composed of a single particle in the presence of an attractive Pöschl-Teller potential in one and three dimensions. The method is general and has potential applicability for large systems.
A rigorous methodology that combines a fully quantum mechanical treatment of a small system with a classical trajectory description of a large number of degrees of freedom is discussed in connection with the mechanism of decoherence. The need for mean field approximations in traditional quantum‐classical calculations is removed by abandoning the delocalized wavefunction description of the quantum particle in favor of Feynman's formulation, which is based on local paths. The effects of the environment enter the quantum‐classical path integral (QCPI) expression in terms of phase factors along classical trajectories, whose number grows exponentially with the number of time steps. It is argued that memory loss allows termination of trajectory branching and that the main contribution to decoherence arises from a single classical trajectory (from each sampled initial condition). Exploiting these ideas allows a dramatic reduction in the required number of trajectories, making QCPI calculations feasible. © 2015 Wiley Periodicals, Inc.
Path integrals are mathematical objects that can be considered as generalizations to an infinite number of variables, represented by paths, of usual integrals. They share the algebraic properties of usual integrals, but have new properties from the viewpoint of analysis. They are powerful tools for the study of quantum mechanics, since they emphasize very explicitly the correspondence between classical and quantum mechanics. Physical quantities are expressed as averages over all possible paths but, in the semi-classical limit, the leading contributions come from paths close to classical paths. Thus, path integrals lead to an intuitive understanding of physical quantities in the semi-classical limit, as well as simple calculations of such quantities. This observation can be illustrated with scattering processes, spectral properties, or barrier penetration effects. Even though the formulation of quantum mechanics based on path integrals seems mathematically more complicated than the usual formulation based on partial differential equations, the path integral formulation is well adapted to systems with many degrees of freedom, where a formalism of Schrödinger type is much less useful. It allows simple construction of a many-body theory both for bosons and fermions.
Recent developments in the chemical literature have shown how discretized path integrals may be used to treat the quantum-mechanical internal degrees of freedom of molecules in liquids. This paper suggests that analogous methods might also be useful for quantum-mechanical solid-state problems, such as those posed by lattice-spin systems. The idea, illustrated in this paper by a common model for tunneling in solids, the transverse Ising model, is to transform the quantal spins into classical spins with ``internal structure.'' The resulting classical system can then be treated by analytical methods. For the transverse Ising model, both the traditional mean-field theory and a new approximation (which includes a fluctuation correction) are derived in this way. The effects of thermal and quantum-mechanical fluctuations are shown to be largely identical for the example considered.